Renyi's (1961) entropy and divergence of order a are given operational characterizations in terms of block coding and hypothesis testing, as so-called β-cutoff rates, with α=(1+β)-1 for entropy and α=(1-β)-1 for divergence. Out of several possible definitions of mutual information and channel capacity of order α, our approach distinguishes one that admits an operational characterization as β-cutoff rate for channel coding, with α=(1-β)-1. The ordinary cutoff rate of a DMC corresponds to β=-1